ECE 6382 Fall 2019 David R Jackson Notes

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ECE 6382 Fall 2019 David R. Jackson Notes 2 Differentiation of Functions of a

ECE 6382 Fall 2019 David R. Jackson Notes 2 Differentiation of Functions of a Complex Variable Notes are adapted from D. R. Wilton, Dept. of ECE 1

Functions of a Complex Variable 2

Functions of a Complex Variable 2

Differentiation of Functions of a Complex Variable 3

Differentiation of Functions of a Complex Variable 3

The Cauchy – Riemann Conditions Denote Augustin-Louis Cauchy Bernhard Riemann 4

The Cauchy – Riemann Conditions Denote Augustin-Louis Cauchy Bernhard Riemann 4

The Cauchy – Riemann Conditions (cont. ) Cauchy-Riemann equations 5

The Cauchy – Riemann Conditions (cont. ) Cauchy-Riemann equations 5

The Cauchy – Riemann Conditions (cont. ) Arbitrary direction z 6

The Cauchy – Riemann Conditions (cont. ) Arbitrary direction z 6

The Cauchy – Riemann Conditions (cont. ) Hence we have the following equivalent statements:

The Cauchy – Riemann Conditions (cont. ) Hence we have the following equivalent statements: 7

The Cauchy – Riemann Conditions (cont. ) D 8

The Cauchy – Riemann Conditions (cont. ) D 8

Applying the Cauchy – Riemann Conditions 9

Applying the Cauchy – Riemann Conditions 9

Applying the Cauchy – Riemann Conditions (cont. ) D A singularity is a point

Applying the Cauchy – Riemann Conditions (cont. ) D A singularity is a point where the function is not analytic. 10

Applying the Cauchy – Riemann Conditions (cont. ) 11

Applying the Cauchy – Riemann Conditions (cont. ) 11

Differentiation Rules (cont. ) Note: The above “brute-force” derivation, directly using the definition of

Differentiation Rules (cont. ) Note: The above “brute-force” derivation, directly using the definition of the derivative, is exactly what is done is usual calculus, with x being used there instead of z. 12

Differentiation Rules 13

Differentiation Rules 13

Differentiation Rules 14

Differentiation Rules 14

A Theorem Related to z* If f = f (z, z*) is analytic, then

A Theorem Related to z* If f = f (z, z*) is analytic, then (The function cannot really vary with z*, and therefore cannot really be a function of z* except in a trivial way. ) 15

A Theorem Related to z* (cont. ) 16

A Theorem Related to z* (cont. ) 16

A Theorem Related to z* (cont. ) 17

A Theorem Related to z* (cont. ) 17

Entire Functions A function that is analytic everywhere in the finite* complex plane is

Entire Functions A function that is analytic everywhere in the finite* complex plane is called “entire”. * A function is said to be analytic everywhere in the finite complex plane if it is analytic everywhere except possibly at infinity. Analytic at infinity: Is the function analytic at w = 0? 18

Combinations of Analytic Functions Combinations of functions: 19

Combinations of Analytic Functions Combinations of functions: 19

Combinations of Analytic Functions (cont. ) Combinations of functions: Example: The first form is

Combinations of Analytic Functions (cont. ) Combinations of functions: Example: The first form is analytic everywhere except z = 1. The second form is analytic for |z| < 1. 20

Combination of Analytic Functions (cont. ) Examples Composite functions of analytic functions are also

Combination of Analytic Functions (cont. ) Examples Composite functions of analytic functions are also analytic Derivatives of analytic functions are also analytic 21

Derivatives of Analytic Function Important theorem (proven later) The derivative of an analytic function

Derivatives of Analytic Function Important theorem (proven later) The derivative of an analytic function is also analytic. Hence, all derivatives of an analytic function are also analytic. 22

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions The functions u and

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions The functions u and v are harmonic (i. e. , they satisfy Laplace’s equation) Notation: This result is extensively used in conformal mapping to solve electrostatics and other problems involving the 2 D Laplace equation (discussed later). Pierre-Simon Laplace 23

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Proof f

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Proof f is analytic df / dz is also analytic (see note on slide 22) 24

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Example: 25

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Example: 25

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Example: 26

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont. ) Example: 26